For all set types S that
are models concept Set (std::set, interval_set,
separate_interval_set
and split_interval_set)
most of the well known mathematical laws
on sets were successfully checked via LaBatea. The next tables are
giving an overview over the checked laws ordered by operations. If possible,
the laws are formulated with the stronger lexicographical equality (operator==)
which implies the law's validity for the weaker element equality is_element_equal. Throughout this chapter
we will denote element equality as =e= instead
of is_element_equal where
a short notation is advantageous.

For set difference there are only these laws. It is not associative and not
commutative. It's neutrality is non symmetrical.

RightNeutrality<S,-,==>:Sa;a-S()==aInversion<S,-,==>:Sa;a-a==S()

Summarized in the next table are laws that use +,
& and -
as a single operation. For all validated laws, the left and right hand sides
of the equations are lexicographically equal, as denoted by == in the cells of the table.

Laws, like distributivity, that use more than one operation can sometimes
be instantiated for different sequences of operators as can be seen below.
In the two instantiations of the distributivity laws operators + and &
are swapped. So we can have small operator signatures like +,& and &,+
to describe such instantiations, which will be used below. Not all instances
of distributivity laws hold for lexicographical equality. Therefore they
are denoted using a variable equality =v=
below.

The table gives an overview over 12 instantiations of the four distributivity
laws and shows the equalities which the instantiations holds for. For instance
RightDistributivity with
operator signature +,- instantiated
for split_interval_sets
holds only for element equality (denoted as =e=):

RightDistributivity<S,+,-,=e=>:Sa,b,c;(a+b)-c=e=(a-c)+(b-c)

The remaining five instantiations of RightDistributivity
are valid for lexicographical equality (demoted as ==)
as well.

De Morgans Law is better known in an incarnation where the unary complement
operation ~ is used. ~(a+b)==~a*~b.
The version below is an adaption for the binary set difference -, which is also called relative complement.